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

Percentage Accurate: 97.0% → 99.0%

Time: 6.9s

Alternatives: 12

Speedup: 0.3×

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: 97.0% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y - z}{\frac{\left(t - z\right) - -1}{a}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-56} \lor \neg \left(t\_1 \leq 5 \cdot 10^{-236}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(y - z, \frac{a}{\mathsf{fma}\left(t - z, x, x\right)}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (- y z) (/ (- (- t z) -1.0) a)))))
   (if (or (<= t_1 -1e-56) (not (<= t_1 5e-236)))
     t_1
     (* (- x) (fma (- y z) (/ a (fma (- t z) x x)) -1.0)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y - z) / (((t - z) - -1.0) / a));
	double tmp;
	if ((t_1 <= -1e-56) || !(t_1 <= 5e-236)) {
		tmp = t_1;
	} else {
		tmp = -x * fma((y - z), (a / fma((t - z), x, x)), -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y - z) / Float64(Float64(Float64(t - z) - -1.0) / a)))
	tmp = 0.0
	if ((t_1 <= -1e-56) || !(t_1 <= 5e-236))
		tmp = t_1;
	else
		tmp = Float64(Float64(-x) * fma(Float64(y - z), Float64(a / fma(Float64(t - z), x, x)), -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] - -1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-56], N[Not[LessEqual[t$95$1, 5e-236]], $MachinePrecision]], t$95$1, N[((-x) * N[(N[(y - z), $MachinePrecision] * N[(a / N[(N[(t - z), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))) < -1e-56 or 4.9999999999999998e-236 < (-.f64 x (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)))

   1. Initial program 99.9%

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

   if -1e-56 < (-.f64 x (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))) < 4.9999999999999998e-236

   1. Initial program 69.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-rgt-neg-out N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{y - z}{\frac{\left(t - z\right) - -1}{a}} \leq -1 \cdot 10^{-56} \lor \neg \left(x - \frac{y - z}{\frac{\left(t - z\right) - -1}{a}} \leq 5 \cdot 10^{-236}\right):\\ \;\;\;\;x - \frac{y - z}{\frac{\left(t - z\right) - -1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(y - z, \frac{a}{\mathsf{fma}\left(t - z, x, x\right)}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) - -1}{a}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-127} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;x - t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (/ (- (- t z) -1.0) a))))
   (if (or (<= t_1 -5e-127) (not (<= t_1 0.0)))
     (- x t_1)
     (fma (/ z (- (+ 1.0 t) z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) - -1.0) / a);
	double tmp;
	if ((t_1 <= -5e-127) || !(t_1 <= 0.0)) {
		tmp = x - t_1;
	} else {
		tmp = fma((z / ((1.0 + t) - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) - -1.0) / a))
	tmp = 0.0
	if ((t_1 <= -5e-127) || !(t_1 <= 0.0))
		tmp = Float64(x - t_1);
	else
		tmp = fma(Float64(z / Float64(Float64(1.0 + t) - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] - -1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e-127], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(x - t$95$1), $MachinePrecision], N[(N[(z / N[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -4.9999999999999997e-127 or -0.0 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.8%

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

   if -4.9999999999999997e-127 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -0.0

   1. Initial program 88.6%

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - z}{\frac{\left(t - z\right) - -1}{a}} \leq -5 \cdot 10^{-127} \lor \neg \left(\frac{y - z}{\frac{\left(t - z\right) - -1}{a}} \leq 0\right):\\ \;\;\;\;x - \frac{y - z}{\frac{\left(t - z\right) - -1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-14} \lor \neg \left(z \leq 3.8 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.4e-14) (not (<= z 3.8e-23)))
   (fma (/ z (- (+ 1.0 t) z)) a x)
   (- x (* (/ y (+ 1.0 t)) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.4e-14) || !(z <= 3.8e-23)) {
		tmp = fma((z / ((1.0 + t) - z)), a, x);
	} else {
		tmp = x - ((y / (1.0 + t)) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.4e-14) || !(z <= 3.8e-23))
		tmp = fma(Float64(z / Float64(Float64(1.0 + t) - z)), a, x);
	else
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 + t)) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.4e-14], N[Not[LessEqual[z, 3.8e-23]], $MachinePrecision]], N[(N[(z / N[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - N[(N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4e-14 or 3.80000000000000011e-23 < z

   1. Initial program 92.6%

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if -1.4e-14 < z < 3.80000000000000011e-23

   1. Initial program 99.1%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 96.1

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

   5. Applied rewrites 96.1%

      \[\leadsto x - \color{blue}{\frac{y}{1 + t} \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-14} \lor \neg \left(z \leq 3.8 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-13}:\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{1 - z}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-23}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.5e-13)
   (- x (* (- y z) (/ a (- 1.0 z))))
   (if (<= z 3.8e-23)
     (- x (* (/ y (+ 1.0 t)) a))
     (fma (/ z (- (+ 1.0 t) z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.5e-13) {
		tmp = x - ((y - z) * (a / (1.0 - z)));
	} else if (z <= 3.8e-23) {
		tmp = x - ((y / (1.0 + t)) * a);
	} else {
		tmp = fma((z / ((1.0 + t) - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.5e-13)
		tmp = Float64(x - Float64(Float64(y - z) * Float64(a / Float64(1.0 - z))));
	elseif (z <= 3.8e-23)
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 + t)) * a));
	else
		tmp = fma(Float64(z / Float64(Float64(1.0 + t) - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.5e-13], N[(x - N[(N[(y - z), $MachinePrecision] * N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e-23], N[(x - N[(N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5000000000000002e-13

   1. Initial program 94.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 91.8

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

   5. Applied rewrites 91.8%

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

   if -3.5000000000000002e-13 < z < 3.80000000000000011e-23

   1. Initial program 99.1%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if 3.80000000000000011e-23 < z

   1. Initial program 91.2%

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 87.0

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

   5. Applied rewrites 87.0%

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

Alternative 5: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-11} \lor \neg \left(z \leq 3.8 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.5e-11) (not (<= z 3.8e-23)))
   (fma (/ z (- t z)) a x)
   (- x (* (/ y (+ 1.0 t)) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.5e-11) || !(z <= 3.8e-23)) {
		tmp = fma((z / (t - z)), a, x);
	} else {
		tmp = x - ((y / (1.0 + t)) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.5e-11) || !(z <= 3.8e-23))
		tmp = fma(Float64(z / Float64(t - z)), a, x);
	else
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 + t)) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.5e-11], N[Not[LessEqual[z, 3.8e-23]], $MachinePrecision]], N[(N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - N[(N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5e-11 or 3.80000000000000011e-23 < z

   1. Initial program 92.6%

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 87.9%

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

   if -1.5e-11 < z < 3.80000000000000011e-23

   1. Initial program 99.1%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 96.1

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

   5. Applied rewrites 96.1%

      \[\leadsto x - \color{blue}{\frac{y}{1 + t} \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-11} \lor \neg \left(z \leq 3.8 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{1 + t} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-38} \lor \neg \left(z \leq 3.8 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y - z\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.3e-38) (not (<= z 3.8e-23)))
   (fma (/ z (- t z)) a x)
   (- x (* (- y z) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.3e-38) || !(z <= 3.8e-23)) {
		tmp = fma((z / (t - z)), a, x);
	} else {
		tmp = x - ((y - z) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.3e-38) || !(z <= 3.8e-23))
		tmp = fma(Float64(z / Float64(t - z)), a, x);
	else
		tmp = Float64(x - Float64(Float64(y - z) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.3e-38], N[Not[LessEqual[z, 3.8e-23]], $MachinePrecision]], N[(N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - N[(N[(y - z), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.30000000000000002e-38 or 3.80000000000000011e-23 < z

   1. Initial program 92.1%

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 87.6%

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

   if -2.30000000000000002e-38 < z < 3.80000000000000011e-23

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto x - \left(y - z\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 81.1%

      \[\leadsto x - \left(y - z\right) \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-38} \lor \neg \left(z \leq 3.8 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y - z\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-35} \lor \neg \left(z \leq 3.8 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y - z\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.1e-35) (not (<= z 3.8e-23)))
   (fma (/ z (- 1.0 z)) a x)
   (- x (* (- y z) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.1e-35) || !(z <= 3.8e-23)) {
		tmp = fma((z / (1.0 - z)), a, x);
	} else {
		tmp = x - ((y - z) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.1e-35) || !(z <= 3.8e-23))
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	else
		tmp = Float64(x - Float64(Float64(y - z) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.1e-35], N[Not[LessEqual[z, 3.8e-23]], $MachinePrecision]], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - N[(N[(y - z), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.10000000000000026e-35 or 3.80000000000000011e-23 < z

   1. Initial program 92.1%

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 89.0

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

   5. Applied rewrites 89.0%

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

   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 79.2%

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

   if -4.10000000000000026e-35 < z < 3.80000000000000011e-23

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto x - \left(y - z\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 81.1%

      \[\leadsto x - \left(y - z\right) \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-35} \lor \neg \left(z \leq 3.8 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y - z\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-35} \lor \neg \left(z \leq 1.7 \cdot 10^{-22}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - \left(y - z\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.4e-35) (not (<= z 1.7e-22))) (- x a) (- x (* (- y z) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.4e-35) || !(z <= 1.7e-22)) {
		tmp = x - a;
	} else {
		tmp = x - ((y - z) * a);
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((z <= (-4.4d-35)) .or. (.not. (z <= 1.7d-22))) then
        tmp = x - a
    else
        tmp = x - ((y - z) * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.4e-35) || !(z <= 1.7e-22)) {
		tmp = x - a;
	} else {
		tmp = x - ((y - z) * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.4e-35) or not (z <= 1.7e-22):
		tmp = x - a
	else:
		tmp = x - ((y - z) * a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.4e-35) || !(z <= 1.7e-22))
		tmp = Float64(x - a);
	else
		tmp = Float64(x - Float64(Float64(y - z) * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -4.4e-35) || ~((z <= 1.7e-22)))
		tmp = x - a;
	else
		tmp = x - ((y - z) * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.4e-35], N[Not[LessEqual[z, 1.7e-22]], $MachinePrecision]], N[(x - a), $MachinePrecision], N[(x - N[(N[(y - z), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.39999999999999987e-35 or 1.6999999999999999e-22 < z

   1. Initial program 92.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 77.4%

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

   if -4.39999999999999987e-35 < z < 1.6999999999999999e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in z around 0

      \[\leadsto x - \left(y - z\right) \cdot a \]
   7. Step-by-step derivation

   8. Applied rewrites 81.4%

      \[\leadsto x - \left(y - z\right) \cdot a \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-35} \lor \neg \left(z \leq 1.7 \cdot 10^{-22}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - \left(y - z\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-69}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-147}:\\ \;\;\;\;\left(-a\right) \cdot y\\ \mathbf{elif}\;z \leq 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.4e-69)
   (- x a)
   (if (<= z -4.1e-147) (* (- a) y) (if (<= z 1e-11) x (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.4e-69) {
		tmp = x - a;
	} else if (z <= -4.1e-147) {
		tmp = -a * y;
	} else if (z <= 1e-11) {
		tmp = x;
	} else {
		tmp = x - a;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (z <= (-6.4d-69)) then
        tmp = x - a
    else if (z <= (-4.1d-147)) then
        tmp = -a * y
    else if (z <= 1d-11) then
        tmp = x
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.4e-69) {
		tmp = x - a;
	} else if (z <= -4.1e-147) {
		tmp = -a * y;
	} else if (z <= 1e-11) {
		tmp = x;
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.4e-69:
		tmp = x - a
	elif z <= -4.1e-147:
		tmp = -a * y
	elif z <= 1e-11:
		tmp = x
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.4e-69)
		tmp = Float64(x - a);
	elseif (z <= -4.1e-147)
		tmp = Float64(Float64(-a) * y);
	elseif (z <= 1e-11)
		tmp = x;
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.4e-69)
		tmp = x - a;
	elseif (z <= -4.1e-147)
		tmp = -a * y;
	elseif (z <= 1e-11)
		tmp = x;
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.4e-69], N[(x - a), $MachinePrecision], If[LessEqual[z, -4.1e-147], N[((-a) * y), $MachinePrecision], If[LessEqual[z, 1e-11], x, N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.39999999999999997e-69 or 9.99999999999999939e-12 < z

   1. Initial program 92.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 76.3%

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

   if -6.39999999999999997e-69 < z < -4.1e-147

   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

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(a \cdot \frac{y}{1 + t}\right) \]

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 72.9

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in t around 0

      \[\leadsto \left(-a\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 59.7%

       \[\leadsto \left(-a\right) \cdot y \]

   if -4.1e-147 < z < 9.99999999999999939e-12

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 67.0%

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

Alternative 10: 72.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-35} \lor \neg \left(z \leq 1.7 \cdot 10^{-22}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.4e-35) (not (<= z 1.7e-22))) (- x a) (- x (* a y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.4e-35) || !(z <= 1.7e-22)) {
		tmp = x - a;
	} else {
		tmp = x - (a * y);
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((z <= (-4.4d-35)) .or. (.not. (z <= 1.7d-22))) then
        tmp = x - a
    else
        tmp = x - (a * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.4e-35) || !(z <= 1.7e-22)) {
		tmp = x - a;
	} else {
		tmp = x - (a * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.4e-35) or not (z <= 1.7e-22):
		tmp = x - a
	else:
		tmp = x - (a * y)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.4e-35) || !(z <= 1.7e-22))
		tmp = Float64(x - a);
	else
		tmp = Float64(x - Float64(a * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -4.4e-35) || ~((z <= 1.7e-22)))
		tmp = x - a;
	else
		tmp = x - (a * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.4e-35], N[Not[LessEqual[z, 1.7e-22]], $MachinePrecision]], N[(x - a), $MachinePrecision], N[(x - N[(a * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.39999999999999987e-35 or 1.6999999999999999e-22 < z

   1. Initial program 92.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 77.4%

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

   if -4.39999999999999987e-35 < z < 1.6999999999999999e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 78.3

         \[\leadsto x - a \cdot y \]

   8. Applied rewrites 78.3%

      \[\leadsto x - a \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-35} \lor \neg \left(z \leq 1.7 \cdot 10^{-22}\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{+190}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+140}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.9e+190) x (if (<= t 6.8e+140) (- x a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.9e+190) {
		tmp = x;
	} else if (t <= 6.8e+140) {
		tmp = x - a;
	} else {
		tmp = x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= (-3.9d+190)) then
        tmp = x
    else if (t <= 6.8d+140) then
        tmp = x - a
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.9e+190) {
		tmp = x;
	} else if (t <= 6.8e+140) {
		tmp = x - a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -3.9e+190:
		tmp = x
	elif t <= 6.8e+140:
		tmp = x - a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3.9e+190)
		tmp = x;
	elseif (t <= 6.8e+140)
		tmp = Float64(x - a);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -3.9e+190)
		tmp = x;
	elseif (t <= 6.8e+140)
		tmp = x - a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3.9e+190], x, If[LessEqual[t, 6.8e+140], N[(x - a), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -3.9000000000000004e190 or 6.8e140 < t

   1. Initial program 93.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 66.5%

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

   if -3.9000000000000004e190 < t < 6.8e140

   1. Initial program 96.5%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 70.6%

      \[\leadsto x - \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 54.0% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 56.9%

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

Developer Target 1: 99.6% 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 2025026 
(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))))