Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Percentage Accurate: 97.1% → 99.7%

Time: 8.4s

Alternatives: 14

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y - z) / (((t - z) + 1.0) / a));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y - z) / (((t - z) + 1.0) / a));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) + 1}, a, x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (fma (/ (- z y) (+ (- t z) 1.0)) a x))

C

double code(double x, double y, double z, double t, double a) {
	return fma(((z - y) / ((t - z) + 1.0)), a, x);
}

Julia

function code(x, y, z, t, a)
	return fma(Float64(Float64(z - y) / Float64(Float64(t - z) + 1.0)), a, x)
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(z - y), $MachinePrecision] / N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]

Derivation

1. Initial program 98.1%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}} \]

   2. lift-/.f64 N/A

      \[\leadsto x - \frac{y - z}{\color{blue}{\frac{\left(t - z\right) + 1}{a}}} \]

   3. associate-/r/ N/A

      \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

   4. lower-*.f64 N/A

      \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

   5. lower-/.f64 99.9

      \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1}} \cdot a \]

   6. lift-+.f64 N/A

      \[\leadsto x - \frac{y - z}{\color{blue}{\left(t - z\right) + 1}} \cdot a \]

   7. +-commutative N/A

      \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

   8. lower-+.f64 99.9

      \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

4. Applied rewrites 99.9%

   \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{x - \frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

   2. lift-*.f64 N/A

      \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a + x} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right), a, x\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{y - z}{1 + \left(t - z\right)}}\right), a, x\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

   8. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

   9. lower-neg.f64 99.9

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-\left(y - z\right)}}{1 + \left(t - z\right)}, a, x\right) \]

   10. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{1 + \left(t - z\right)}}, a, x\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

   12. lower-+.f64 99.9

       \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(y - z\right)}{\left(t - z\right) + 1}, a, x\right)} \]

7. Taylor expanded in z around 0

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\left(t - z\right) + 1}, a, x\right) \]
8. Step-by-step derivation

   1. lower--.f64 99.9

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\left(t - z\right) + 1}, a, x\right) \]

9. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\left(t - z\right) + 1}, a, x\right) \]
10. Add Preprocessing

Alternative 2: 76.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -7.8 \cdot 10^{-130}:\\ \;\;\;\;x - \frac{y}{1 - z} \cdot a\\ \mathbf{elif}\;t \leq 105000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) t) a x)))
   (if (<= t -3.8e+64)
     t_1
     (if (<= t -7.8e-130)
       (- x (* (/ y (- 1.0 z)) a))
       (if (<= t 105000000000.0) (fma (/ z (- 1.0 z)) a x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / t), a, x);
	double tmp;
	if (t <= -3.8e+64) {
		tmp = t_1;
	} else if (t <= -7.8e-130) {
		tmp = x - ((y / (1.0 - z)) * a);
	} else if (t <= 105000000000.0) {
		tmp = fma((z / (1.0 - z)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / t), a, x)
	tmp = 0.0
	if (t <= -3.8e+64)
		tmp = t_1;
	elseif (t <= -7.8e-130)
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 - z)) * a));
	elseif (t <= 105000000000.0)
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[t, -3.8e+64], t$95$1, If[LessEqual[t, -7.8e-130], N[(x - N[(N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 105000000000.0], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.8000000000000001e64 or 1.05e11 < t

   1. Initial program 98.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{\left(t - z\right) + 1}{a}}} \]

      3. associate-/r/ N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      5. lower-/.f64 99.9

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1}} \cdot a \]

      6. lift-+.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\left(t - z\right) + 1}} \cdot a \]

      7. +-commutative N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

      8. lower-+.f64 99.9

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

   4. Applied rewrites 99.9%

      \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a + x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right), a, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{y - z}{1 + \left(t - z\right)}}\right), a, x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      9. lower-neg.f64 99.9

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-\left(y - z\right)}}{1 + \left(t - z\right)}, a, x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{1 + \left(t - z\right)}}, a, x\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

      12. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(y - z\right)}{\left(t - z\right) + 1}, a, x\right)} \]

   7. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{t}}, a, x\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{t}}, a, x\right) \]

      2. lower--.f64 89.6

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{t}, a, x\right) \]

   9. Applied rewrites 89.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{t}}, a, x\right) \]

   if -3.8000000000000001e64 < t < -7.8000000000000002e-130

   1. Initial program 100.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x - \color{blue}{\frac{a \cdot y}{\left(1 + t\right) - z}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{a \cdot \frac{y}{\left(1 + t\right) - z}} \]

      2. *-commutative N/A

         \[\leadsto x - \color{blue}{\frac{y}{\left(1 + t\right) - z} \cdot a} \]

      3. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y}{\left(1 + t\right) - z} \cdot a} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y}{\left(1 + t\right) - z}} \cdot a \]

      5. lower--.f64 N/A

         \[\leadsto x - \frac{y}{\color{blue}{\left(1 + t\right) - z}} \cdot a \]

      6. lower-+.f64 89.9

         \[\leadsto x - \frac{y}{\color{blue}{\left(1 + t\right)} - z} \cdot a \]

   5. Applied rewrites 89.9%

      \[\leadsto x - \color{blue}{\frac{y}{\left(1 + t\right) - z} \cdot a} \]

   6. Taylor expanded in t around 0

      \[\leadsto x - \frac{y}{1 - z} \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 84.6%

      \[\leadsto x - \frac{y}{1 - z} \cdot a \]

   if -7.8000000000000002e-130 < t < 1.05e11

   1. Initial program 97.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x} \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x \]

      4. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(1 + t\right) - z}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{z}{\left(1 + t\right) - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{\left(1 + t\right) - z} \cdot a} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{\left(1 + t\right) - z}}, a, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right) - z}}, a, x\right) \]

      10. lower-+.f64 77.7

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right)} - z}, a, x\right) \]

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.7%

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{t} \cdot a\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - y, z, -y\right), a, x\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* (/ y t) a))))
   (if (<= t -4.8e-59)
     t_1
     (if (<= t -8e-106)
       (fma (fma (- 1.0 y) z (- y)) a x)
       (if (<= t 5e+33) (fma (/ z (- 1.0 z)) a x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y / t) * a);
	double tmp;
	if (t <= -4.8e-59) {
		tmp = t_1;
	} else if (t <= -8e-106) {
		tmp = fma(fma((1.0 - y), z, -y), a, x);
	} else if (t <= 5e+33) {
		tmp = fma((z / (1.0 - z)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y / t) * a))
	tmp = 0.0
	if (t <= -4.8e-59)
		tmp = t_1;
	elseif (t <= -8e-106)
		tmp = fma(fma(Float64(1.0 - y), z, Float64(-y)), a, x);
	elseif (t <= 5e+33)
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y / t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.8e-59], t$95$1, If[LessEqual[t, -8e-106], N[(N[(N[(1.0 - y), $MachinePrecision] * z + (-y)), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[t, 5e+33], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.8000000000000003e-59 or 4.99999999999999973e33 < t

   1. Initial program 98.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x - \color{blue}{\frac{a \cdot y}{\left(1 + t\right) - z}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{a \cdot \frac{y}{\left(1 + t\right) - z}} \]

      2. *-commutative N/A

         \[\leadsto x - \color{blue}{\frac{y}{\left(1 + t\right) - z} \cdot a} \]

      3. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y}{\left(1 + t\right) - z} \cdot a} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y}{\left(1 + t\right) - z}} \cdot a \]

      5. lower--.f64 N/A

         \[\leadsto x - \frac{y}{\color{blue}{\left(1 + t\right) - z}} \cdot a \]

      6. lower-+.f64 88.2

         \[\leadsto x - \frac{y}{\color{blue}{\left(1 + t\right)} - z} \cdot a \]

   5. Applied rewrites 88.2%

      \[\leadsto x - \color{blue}{\frac{y}{\left(1 + t\right) - z} \cdot a} \]

   6. Taylor expanded in t around inf

      \[\leadsto x - \frac{y}{t} \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 85.7%

      \[\leadsto x - \frac{y}{t} \cdot a \]

   if -4.8000000000000003e-59 < t < -7.99999999999999953e-106

   1. Initial program 99.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{\left(t - z\right) + 1}{a}}} \]

      3. associate-/r/ N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      5. lower-/.f64 100.0

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1}} \cdot a \]

      6. lift-+.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\left(t - z\right) + 1}} \cdot a \]

      7. +-commutative N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

      8. lower-+.f64 100.0

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

   4. Applied rewrites 100.0%

      \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a + x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right), a, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{y - z}{1 + \left(t - z\right)}}\right), a, x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      9. lower-neg.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-\left(y - z\right)}}{1 + \left(t - z\right)}, a, x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{1 + \left(t - z\right)}}, a, x\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

      12. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(y - z\right)}{\left(t - z\right) + 1}, a, x\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{1 - z}}, a, x\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{1 - z}}, a, x\right) \]

      2. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{1 - z}, a, x\right) \]

      3. lower--.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{1 - z}}, a, x\right) \]

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{1 - z}}, a, x\right) \]

   10. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(-1 \cdot y + \color{blue}{z \cdot \left(1 - y\right)}, a, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 92.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(1 - y, \color{blue}{z}, -y\right), a, x\right) \]

   if -7.99999999999999953e-106 < t < 4.99999999999999973e33

   1. Initial program 97.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x} \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x \]

      4. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(1 + t\right) - z}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{z}{\left(1 + t\right) - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{\left(1 + t\right) - z} \cdot a} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{\left(1 + t\right) - z}}, a, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right) - z}}, a, x\right) \]

      10. lower-+.f64 77.8

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right)} - z}, a, x\right) \]

   5. Applied rewrites 77.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.1%

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+65} \lor \neg \left(t \leq 190000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -3.4e+65) (not (<= t 190000000000.0)))
   (fma (/ (- z y) t) a x)
   (fma (/ (- z y) (- 1.0 z)) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.4e+65) || !(t <= 190000000000.0)) {
		tmp = fma(((z - y) / t), a, x);
	} else {
		tmp = fma(((z - y) / (1.0 - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -3.4e+65) || !(t <= 190000000000.0))
		tmp = fma(Float64(Float64(z - y) / t), a, x);
	else
		tmp = fma(Float64(Float64(z - y) / Float64(1.0 - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -3.4e+65], N[Not[LessEqual[t, 190000000000.0]], $MachinePrecision]], N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.3999999999999999e65 or 1.9e11 < t

   1. Initial program 98.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{\left(t - z\right) + 1}{a}}} \]

      3. associate-/r/ N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      5. lower-/.f64 99.9

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1}} \cdot a \]

      6. lift-+.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\left(t - z\right) + 1}} \cdot a \]

      7. +-commutative N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

      8. lower-+.f64 99.9

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

   4. Applied rewrites 99.9%

      \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a + x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right), a, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{y - z}{1 + \left(t - z\right)}}\right), a, x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      9. lower-neg.f64 99.9

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-\left(y - z\right)}}{1 + \left(t - z\right)}, a, x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{1 + \left(t - z\right)}}, a, x\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

      12. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(y - z\right)}{\left(t - z\right) + 1}, a, x\right)} \]

   7. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{t}}, a, x\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{t}}, a, x\right) \]

      2. lower--.f64 90.4

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{t}, a, x\right) \]

   9. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{t}}, a, x\right) \]

   if -3.3999999999999999e65 < t < 1.9e11

   1. Initial program 98.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{\left(t - z\right) + 1}{a}}} \]

      3. associate-/r/ N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      5. lower-/.f64 99.9

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1}} \cdot a \]

      6. lift-+.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\left(t - z\right) + 1}} \cdot a \]

      7. +-commutative N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

      8. lower-+.f64 99.9

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

   4. Applied rewrites 99.9%

      \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a + x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right), a, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{y - z}{1 + \left(t - z\right)}}\right), a, x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      9. lower-neg.f64 99.9

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-\left(y - z\right)}}{1 + \left(t - z\right)}, a, x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{1 + \left(t - z\right)}}, a, x\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

      12. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(y - z\right)}{\left(t - z\right) + 1}, a, x\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{1 - z}}, a, x\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{1 - z}}, a, x\right) \]

      2. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{1 - z}, a, x\right) \]

      3. lower--.f64 98.1

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{1 - z}}, a, x\right) \]

   9. Applied rewrites 98.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{1 - z}}, a, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+65} \lor \neg \left(t \leq 190000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.00041 \lor \neg \left(z \leq 2.9 \cdot 10^{-15}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -0.00041) (not (<= z 2.9e-15)))
   (fma (/ z (- (+ 1.0 t) z)) a x)
   (fma (/ y (- -1.0 t)) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -0.00041) || !(z <= 2.9e-15)) {
		tmp = fma((z / ((1.0 + t) - z)), a, x);
	} else {
		tmp = fma((y / (-1.0 - t)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -0.00041) || !(z <= 2.9e-15))
		tmp = fma(Float64(z / Float64(Float64(1.0 + t) - z)), a, x);
	else
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -0.00041], N[Not[LessEqual[z, 2.9e-15]], $MachinePrecision]], N[(N[(z / N[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.0999999999999999e-4 or 2.90000000000000019e-15 < z

   1. Initial program 97.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x} \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x \]

      4. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(1 + t\right) - z}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{z}{\left(1 + t\right) - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{\left(1 + t\right) - z} \cdot a} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{\left(1 + t\right) - z}}, a, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right) - z}}, a, x\right) \]

      10. lower-+.f64 89.1

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right)} - z}, a, x\right) \]

   5. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

   if -4.0999999999999999e-4 < z < 2.90000000000000019e-15

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{\left(t - z\right) + 1}{a}}} \]

      3. associate-/r/ N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      5. lower-/.f64 99.9

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1}} \cdot a \]

      6. lift-+.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\left(t - z\right) + 1}} \cdot a \]

      7. +-commutative N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

      8. lower-+.f64 99.9

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

   4. Applied rewrites 99.9%

      \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a + x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right), a, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{y - z}{1 + \left(t - z\right)}}\right), a, x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      9. lower-neg.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-\left(y - z\right)}}{1 + \left(t - z\right)}, a, x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{1 + \left(t - z\right)}}, a, x\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

      12. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(y - z\right)}{\left(t - z\right) + 1}, a, x\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \frac{y}{1 + t}}, a, x\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot y}{1 + t}}, a, x\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot y}{1 + t}}, a, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(y\right)}}{1 + t}, a, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-y}}{1 + t}, a, x\right) \]

      5. lower-+.f64 93.4

         \[\leadsto \mathsf{fma}\left(\frac{-y}{\color{blue}{1 + t}}, a, x\right) \]

   9. Applied rewrites 93.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-y}{1 + t}}, a, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.00041 \lor \neg \left(z \leq 2.9 \cdot 10^{-15}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.00041 \lor \neg \left(z \leq 2.9 \cdot 10^{-15}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{\left(t - z\right) + 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -0.00041) (not (<= z 2.9e-15)))
   (fma z (/ a (+ (- t z) 1.0)) x)
   (fma (/ y (- -1.0 t)) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -0.00041) || !(z <= 2.9e-15)) {
		tmp = fma(z, (a / ((t - z) + 1.0)), x);
	} else {
		tmp = fma((y / (-1.0 - t)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -0.00041) || !(z <= 2.9e-15))
		tmp = fma(z, Float64(a / Float64(Float64(t - z) + 1.0)), x);
	else
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -0.00041], N[Not[LessEqual[z, 2.9e-15]], $MachinePrecision]], N[(z * N[(a / N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.0999999999999999e-4 or 2.90000000000000019e-15 < z

   1. Initial program 97.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x} \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x \]

      4. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(1 + t\right) - z}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{z}{\left(1 + t\right) - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{\left(1 + t\right) - z} \cdot a} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{\left(1 + t\right) - z}}, a, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right) - z}}, a, x\right) \]

      10. lower-+.f64 89.1

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right)} - z}, a, x\right) \]

   5. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.4%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{a}{\left(t - z\right) + 1}}, x\right) \]

   if -4.0999999999999999e-4 < z < 2.90000000000000019e-15

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{\left(t - z\right) + 1}{a}}} \]

      3. associate-/r/ N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      5. lower-/.f64 99.9

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1}} \cdot a \]

      6. lift-+.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\left(t - z\right) + 1}} \cdot a \]

      7. +-commutative N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

      8. lower-+.f64 99.9

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

   4. Applied rewrites 99.9%

      \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a + x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right), a, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{y - z}{1 + \left(t - z\right)}}\right), a, x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      9. lower-neg.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-\left(y - z\right)}}{1 + \left(t - z\right)}, a, x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{1 + \left(t - z\right)}}, a, x\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

      12. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(y - z\right)}{\left(t - z\right) + 1}, a, x\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \frac{y}{1 + t}}, a, x\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot y}{1 + t}}, a, x\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot y}{1 + t}}, a, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(y\right)}}{1 + t}, a, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-y}}{1 + t}, a, x\right) \]

      5. lower-+.f64 93.4

         \[\leadsto \mathsf{fma}\left(\frac{-y}{\color{blue}{1 + t}}, a, x\right) \]

   9. Applied rewrites 93.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-y}{1 + t}}, a, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.00041 \lor \neg \left(z \leq 2.9 \cdot 10^{-15}\right):\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{\left(t - z\right) + 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0075:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -0.0075)
   (fma (/ z (- 1.0 z)) a x)
   (if (<= z 7.6e+18) (- x (* (/ y (+ 1.0 t)) a)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.0075) {
		tmp = fma((z / (1.0 - z)), a, x);
	} else if (z <= 7.6e+18) {
		tmp = x - ((y / (1.0 + t)) * a);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -0.0075)
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	elseif (z <= 7.6e+18)
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 + t)) * a));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -0.0075], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 7.6e+18], N[(x - N[(N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.0074999999999999997

   1. Initial program 97.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x} \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x \]

      4. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(1 + t\right) - z}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{z}{\left(1 + t\right) - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{\left(1 + t\right) - z} \cdot a} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{\left(1 + t\right) - z}}, a, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right) - z}}, a, x\right) \]

      10. lower-+.f64 88.1

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right)} - z}, a, x\right) \]

   5. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 84.7%

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]

   if -0.0074999999999999997 < z < 7.6e18

   1. Initial program 99.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x - \color{blue}{\frac{a \cdot y}{1 + t}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - \color{blue}{a \cdot \frac{y}{1 + t}} \]

      2. *-commutative N/A

         \[\leadsto x - \color{blue}{\frac{y}{1 + t} \cdot a} \]

      3. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y}{1 + t} \cdot a} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y}{1 + t}} \cdot a \]

      5. lower-+.f64 91.7

         \[\leadsto x - \frac{y}{\color{blue}{1 + t}} \cdot a \]

   5. Applied rewrites 91.7%

      \[\leadsto x - \color{blue}{\frac{y}{1 + t} \cdot a} \]

   if 7.6e18 < z

   1. Initial program 96.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x - a} \]
   4. Step-by-step derivation

      1. lower--.f64 83.3

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 83.3%

      \[\leadsto \color{blue}{x - a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+65} \lor \neg \left(t \leq 105000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -3.6e+65) (not (<= t 105000000000.0)))
   (fma (/ (- z y) t) a x)
   (fma (/ z (- 1.0 z)) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.6e+65) || !(t <= 105000000000.0)) {
		tmp = fma(((z - y) / t), a, x);
	} else {
		tmp = fma((z / (1.0 - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -3.6e+65) || !(t <= 105000000000.0))
		tmp = fma(Float64(Float64(z - y) / t), a, x);
	else
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -3.6e+65], N[Not[LessEqual[t, 105000000000.0]], $MachinePrecision]], N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.59999999999999978e65 or 1.05e11 < t

   1. Initial program 98.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{\left(t - z\right) + 1}{a}}} \]

      3. associate-/r/ N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      5. lower-/.f64 99.9

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1}} \cdot a \]

      6. lift-+.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\left(t - z\right) + 1}} \cdot a \]

      7. +-commutative N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

      8. lower-+.f64 99.9

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

   4. Applied rewrites 99.9%

      \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a + x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right), a, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{y - z}{1 + \left(t - z\right)}}\right), a, x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      9. lower-neg.f64 99.9

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-\left(y - z\right)}}{1 + \left(t - z\right)}, a, x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{1 + \left(t - z\right)}}, a, x\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

      12. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(y - z\right)}{\left(t - z\right) + 1}, a, x\right)} \]

   7. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{t}}, a, x\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{t}}, a, x\right) \]

      2. lower--.f64 89.6

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{t}, a, x\right) \]

   9. Applied rewrites 89.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{t}}, a, x\right) \]

   if -3.59999999999999978e65 < t < 1.05e11

   1. Initial program 98.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x} \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x \]

      4. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(1 + t\right) - z}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{z}{\left(1 + t\right) - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{\left(1 + t\right) - z} \cdot a} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{\left(1 + t\right) - z}}, a, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right) - z}}, a, x\right) \]

      10. lower-+.f64 76.5

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right)} - z}, a, x\right) \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.0%

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+65} \lor \neg \left(t \leq 105000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0075:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -0.0075)
   (fma (/ z (- 1.0 z)) a x)
   (if (<= z 7.6e+18) (fma (/ y (- -1.0 t)) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.0075) {
		tmp = fma((z / (1.0 - z)), a, x);
	} else if (z <= 7.6e+18) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -0.0075)
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	elseif (z <= 7.6e+18)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -0.0075], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 7.6e+18], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.0074999999999999997

   1. Initial program 97.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x} \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x \]

      4. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(1 + t\right) - z}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{z}{\left(1 + t\right) - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{\left(1 + t\right) - z} \cdot a} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{\left(1 + t\right) - z}}, a, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right) - z}}, a, x\right) \]

      10. lower-+.f64 88.1

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right)} - z}, a, x\right) \]

   5. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 84.7%

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]

   if -0.0074999999999999997 < z < 7.6e18

   1. Initial program 99.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{\left(t - z\right) + 1}{a}}} \]

      3. associate-/r/ N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      5. lower-/.f64 99.9

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1}} \cdot a \]

      6. lift-+.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\left(t - z\right) + 1}} \cdot a \]

      7. +-commutative N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

      8. lower-+.f64 99.9

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

   4. Applied rewrites 99.9%

      \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a + x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right), a, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{y - z}{1 + \left(t - z\right)}}\right), a, x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      9. lower-neg.f64 99.9

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-\left(y - z\right)}}{1 + \left(t - z\right)}, a, x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{1 + \left(t - z\right)}}, a, x\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

      12. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(y - z\right)}{\left(t - z\right) + 1}, a, x\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \frac{y}{1 + t}}, a, x\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot y}{1 + t}}, a, x\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot y}{1 + t}}, a, x\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(y\right)}}{1 + t}, a, x\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-y}}{1 + t}, a, x\right) \]

      5. lower-+.f64 91.7

         \[\leadsto \mathsf{fma}\left(\frac{-y}{\color{blue}{1 + t}}, a, x\right) \]

   9. Applied rewrites 91.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-y}{1 + t}}, a, x\right) \]

   if 7.6e18 < z

   1. Initial program 96.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x - a} \]
   4. Step-by-step derivation

      1. lower--.f64 83.3

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 83.3%

      \[\leadsto \color{blue}{x - a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0075:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.98 \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - y, z, -y\right), a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -0.98) (not (<= z 8e-6)))
   (- x a)
   (fma (fma (- 1.0 y) z (- y)) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -0.98) || !(z <= 8e-6)) {
		tmp = x - a;
	} else {
		tmp = fma(fma((1.0 - y), z, -y), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -0.98) || !(z <= 8e-6))
		tmp = Float64(x - a);
	else
		tmp = fma(fma(Float64(1.0 - y), z, Float64(-y)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -0.98], N[Not[LessEqual[z, 8e-6]], $MachinePrecision]], N[(x - a), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] * z + (-y)), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.97999999999999998 or 7.99999999999999964e-6 < z

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x - a} \]
   4. Step-by-step derivation

      1. lower--.f64 82.2

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 82.2%

      \[\leadsto \color{blue}{x - a} \]

   if -0.97999999999999998 < z < 7.99999999999999964e-6

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\frac{\left(t - z\right) + 1}{a}}} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{\left(t - z\right) + 1}{a}}} \]

      3. associate-/r/ N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      4. lower-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

      5. lower-/.f64 99.9

         \[\leadsto x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1}} \cdot a \]

      6. lift-+.f64 N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\left(t - z\right) + 1}} \cdot a \]

      7. +-commutative N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

      8. lower-+.f64 99.9

         \[\leadsto x - \frac{y - z}{\color{blue}{1 + \left(t - z\right)}} \cdot a \]

   4. Applied rewrites 99.9%

      \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y - z}{1 + \left(t - z\right)} \cdot a} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right)\right) \cdot a + x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{y - z}{1 + \left(t - z\right)}\right), a, x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{y - z}{1 + \left(t - z\right)}}\right), a, x\right) \]

      7. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{1 + \left(t - z\right)}}, a, x\right) \]

      9. lower-neg.f64 99.9

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-\left(y - z\right)}}{1 + \left(t - z\right)}, a, x\right) \]

      10. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{1 + \left(t - z\right)}}, a, x\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

      12. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(\frac{-\left(y - z\right)}{\color{blue}{\left(t - z\right) + 1}}, a, x\right) \]

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\left(y - z\right)}{\left(t - z\right) + 1}, a, x\right)} \]

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{1 - z}}, a, x\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{1 - z}}, a, x\right) \]

      2. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{1 - z}, a, x\right) \]

      3. lower--.f64 73.9

         \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{1 - z}}, a, x\right) \]

   9. Applied rewrites 73.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{1 - z}}, a, x\right) \]

   10. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(-1 \cdot y + \color{blue}{z \cdot \left(1 - y\right)}, a, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 73.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(1 - y, \color{blue}{z}, -y\right), a, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.98 \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - y, z, -y\right), a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+14} \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.4e+14) (not (<= z 8e-6))) (- x a) (fma (/ z 1.0) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.4e+14) || !(z <= 8e-6)) {
		tmp = x - a;
	} else {
		tmp = fma((z / 1.0), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.4e+14) || !(z <= 8e-6))
		tmp = Float64(x - a);
	else
		tmp = fma(Float64(z / 1.0), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.4e+14], N[Not[LessEqual[z, 8e-6]], $MachinePrecision]], N[(x - a), $MachinePrecision], N[(N[(z / 1.0), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4e14 or 7.99999999999999964e-6 < z

   1. Initial program 96.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x - a} \]
   4. Step-by-step derivation

      1. lower--.f64 82.9

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 82.9%

      \[\leadsto \color{blue}{x - a} \]

   if -2.4e14 < z < 7.99999999999999964e-6

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x} \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x \]

      4. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(1 + t\right) - z}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{z}{\left(1 + t\right) - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{\left(1 + t\right) - z} \cdot a} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{\left(1 + t\right) - z}}, a, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right) - z}}, a, x\right) \]

      10. lower-+.f64 66.0

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right)} - z}, a, x\right) \]

   5. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 61.6%

      \[\leadsto \mathsf{fma}\left(\frac{z}{1 - z}, a, x\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{1}, a, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 61.1%

       \[\leadsto \mathsf{fma}\left(\frac{z}{1}, a, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+14} \lor \neg \left(z \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-27} \lor \neg \left(z \leq 1.65 \cdot 10^{-48}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.4e-27) (not (<= z 1.65e-48))) (- x a) (fma (/ z t) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.4e-27) || !(z <= 1.65e-48)) {
		tmp = x - a;
	} else {
		tmp = fma((z / t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.4e-27) || !(z <= 1.65e-48))
		tmp = Float64(x - a);
	else
		tmp = fma(Float64(z / t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.4e-27], N[Not[LessEqual[z, 1.65e-48]], $MachinePrecision]], N[(x - a), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3999999999999997e-27 or 1.65e-48 < z

   1. Initial program 97.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x - a} \]
   4. Step-by-step derivation

      1. lower--.f64 75.8

         \[\leadsto \color{blue}{x - a} \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{x - a} \]

   if -3.3999999999999997e-27 < z < 1.65e-48

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x} \]

      3. metadata-eval N/A

         \[\leadsto \color{blue}{1} \cdot \frac{a \cdot z}{\left(1 + t\right) - z} + x \]

      4. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{a \cdot z}{\left(1 + t\right) - z}} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{z}{\left(1 + t\right) - z}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{\left(1 + t\right) - z} \cdot a} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{\left(1 + t\right) - z}}, a, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right) - z}}, a, x\right) \]

      10. lower-+.f64 69.0

          \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{\left(1 + t\right)} - z}, a, x\right) \]

   5. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 60.1%

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, a, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-27} \lor \neg \left(z \leq 1.65 \cdot 10^{-48}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ x - a \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- x a))

C

double code(double x, double y, double z, double t, double a) {
	return x - a;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - a
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - a;
}

Python

def code(x, y, z, t, a):
	return x - a

Julia

function code(x, y, z, t, a)
	return Float64(x - a)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - a;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - a), $MachinePrecision]

Derivation

1. Initial program 98.1%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x - a} \]
4. Step-by-step derivation

   1. lower--.f64 61.8

      \[\leadsto \color{blue}{x - a} \]

5. Applied rewrites 61.8%

   \[\leadsto \color{blue}{x - a} \]
6. Add Preprocessing

Alternative 14: 16.9% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ -a \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- a))

C

double code(double x, double y, double z, double t, double a) {
	return -a;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = -a
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -a;
}

Python

def code(x, y, z, t, a):
	return -a

Julia

function code(x, y, z, t, a)
	return Float64(-a)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -a;
end

Wolfram

code[x_, y_, z_, t_, a_] := (-a)

Derivation

1. Initial program 98.1%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x - a} \]
4. Step-by-step derivation

   1. lower--.f64 61.8

      \[\leadsto \color{blue}{x - a} \]

5. Applied rewrites 61.8%

   \[\leadsto \color{blue}{x - a} \]

6. Taylor expanded in x around 0

   \[\leadsto -1 \cdot \color{blue}{a} \]
7. Step-by-step derivation

8. Applied rewrites 17.2%

   \[\leadsto -a \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ x - \frac{y - z}{\left(t - z\right) + 1} \cdot a \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- x (* (/ (- y z) (+ (- t z) 1.0)) a)))

C

double code(double x, double y, double z, double t, double a) {
	return x - (((y - z) / ((t - z) + 1.0)) * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - (((y - z) / ((t - z) + 1.0d0)) * a)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - (((y - z) / ((t - z) + 1.0)) * a);
}

Python

def code(x, y, z, t, a):
	return x - (((y - z) / ((t - z) + 1.0)) * a)

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(Float64(y - z) / Float64(Float64(t - z) + 1.0)) * a))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - (((y - z) / ((t - z) + 1.0)) * a);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(N[(y - z), $MachinePrecision] / N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024363 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (/ (- y z) (+ (- t z) 1)) a)))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))