Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 2.7s

Alternatives: 10

Speedup: 1.1×

• 65.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[x + \left(y - z\right) \cdot \left(t - x\right) \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - z) * (t - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[x + \left(y - z\right) \cdot \left(t - x\right) \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - z) * (t - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

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

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

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

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 100.0%

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

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z - y, x - t, x\right)} \]
4. Add Preprocessing

Alternative 2: 81.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y - z, t, x\right)\\ \mathbf{if}\;t \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(z - y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- y z) t x)))
   (if (<= t -3.45e+46) t_1 (if (<= t 5.1e-30) (fma (- z y) x x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y - z), t, x);
	double tmp;
	if (t <= -3.45e+46) {
		tmp = t_1;
	} else if (t <= 5.1e-30) {
		tmp = fma((z - y), x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y - z), t, x)
	tmp = 0.0
	if (t <= -3.45e+46)
		tmp = t_1;
	elseif (t <= 5.1e-30)
		tmp = fma(Float64(z - y), x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[t, -3.45e+46], t$95$1, If[LessEqual[t, 5.1e-30], N[(N[(z - y), $MachinePrecision] * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.4500000000000001e46 or 5.0999999999999997e-30 < t

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 64.8%

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

   6. Applied rewrites 64.8%

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

   if -3.4500000000000001e46 < t < 5.0999999999999997e-30

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 56.3%

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

   6. Applied rewrites 56.3%

      \[\leadsto \color{blue}{x + x \cdot \left(z - y\right)} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sub-flip N/A

         \[\leadsto x + x \cdot \left(z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto x + x \cdot \left(\left(\mathsf{neg}\left(y\right)\right) + \color{blue}{z}\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto x + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot x + \color{blue}{z \cdot x}\right) \]

      6. *-commutative N/A

         \[\leadsto x + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot x + x \cdot \color{blue}{z}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot x + x \cdot \color{blue}{z}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\mathsf{neg}\left(y\right), \color{blue}{x}, x \cdot z\right) \]

      9. lower-neg.f64 55.4%

         \[\leadsto x + \mathsf{fma}\left(-y, x, x \cdot z\right) \]

      10. lift-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(-y, x, x \cdot z\right) \]

      11. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(-y, x, z \cdot x\right) \]

      12. lower-*.f64 55.4%

          \[\leadsto x + \mathsf{fma}\left(-y, x, z \cdot x\right) \]

   8. Applied rewrites 55.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

         \[\leadsto \left(\left(-y\right) \cdot x + z \cdot x\right) + x \]

      4. +-commutative N/A

         \[\leadsto \left(z \cdot x + \left(-y\right) \cdot x\right) + x \]

      5. lift-*.f64 N/A

         \[\leadsto \left(z \cdot x + \left(-y\right) \cdot x\right) + x \]

      6. distribute-rgt-out N/A

         \[\leadsto x \cdot \left(z + \left(-y\right)\right) + x \]

      7. lift-neg.f64 N/A

         \[\leadsto x \cdot \left(z + \left(\mathsf{neg}\left(y\right)\right)\right) + x \]

      8. sub-flip N/A

         \[\leadsto x \cdot \left(z - y\right) + x \]

      9. *-commutative N/A

         \[\leadsto \left(z - y\right) \cdot x + x \]

      10. lower-fma.f64 N/A

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

      11. lower--.f64 56.3%

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

   10. Applied rewrites 56.3%

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

Alternative 3: 81.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 420000000000:\\ \;\;\;\;\mathsf{fma}\left(z, x - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- t x))))
   (if (<= y -2.1e+139) t_1 (if (<= y 420000000000.0) (fma z (- x t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double tmp;
	if (y <= -2.1e+139) {
		tmp = t_1;
	} else if (y <= 420000000000.0) {
		tmp = fma(z, (x - t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	tmp = 0.0
	if (y <= -2.1e+139)
		tmp = t_1;
	elseif (y <= 420000000000.0)
		tmp = fma(z, Float64(x - t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1e+139], t$95$1, If[LessEqual[y, 420000000000.0], N[(z * N[(x - t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0999999999999999e139 or 4.2e11 < y

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 64.8%

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(t - x\right)} \]

      2. lower--.f64 44.7%

         \[\leadsto y \cdot \left(t - \color{blue}{x}\right) \]

   9. Applied rewrites 44.7%

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]

   if -2.0999999999999999e139 < y < 4.2e11

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 60.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z}, x - t, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := t \cdot \left(y - z\right)\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(z - y, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (- y z))))
   (if (<= t -4.4e+92) t_1 (if (<= t 4.4e+42) (fma (- z y) x x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t * (y - z);
	double tmp;
	if (t <= -4.4e+92) {
		tmp = t_1;
	} else if (t <= 4.4e+42) {
		tmp = fma((z - y), x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(t * Float64(y - z))
	tmp = 0.0
	if (t <= -4.4e+92)
		tmp = t_1;
	elseif (t <= 4.4e+42)
		tmp = fma(Float64(z - y), x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.4e+92], t$95$1, If[LessEqual[t, 4.4e+42], N[(N[(z - y), $MachinePrecision] * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.3999999999999998e92 or 4.4000000000000003e42 < t

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 64.8%

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{t \cdot \left(y - z\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 49.3%

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

   9. Applied rewrites 49.3%

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

   if -4.3999999999999998e92 < t < 4.4000000000000003e42

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 56.3%

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

   6. Applied rewrites 56.3%

      \[\leadsto \color{blue}{x + x \cdot \left(z - y\right)} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sub-flip N/A

         \[\leadsto x + x \cdot \left(z + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto x + x \cdot \left(\left(\mathsf{neg}\left(y\right)\right) + \color{blue}{z}\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto x + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot x + \color{blue}{z \cdot x}\right) \]

      6. *-commutative N/A

         \[\leadsto x + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot x + x \cdot \color{blue}{z}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x + \left(\left(\mathsf{neg}\left(y\right)\right) \cdot x + x \cdot \color{blue}{z}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(\mathsf{neg}\left(y\right), \color{blue}{x}, x \cdot z\right) \]

      9. lower-neg.f64 55.4%

         \[\leadsto x + \mathsf{fma}\left(-y, x, x \cdot z\right) \]

      10. lift-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(-y, x, x \cdot z\right) \]

      11. *-commutative N/A

          \[\leadsto x + \mathsf{fma}\left(-y, x, z \cdot x\right) \]

      12. lower-*.f64 55.4%

          \[\leadsto x + \mathsf{fma}\left(-y, x, z \cdot x\right) \]

   8. Applied rewrites 55.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

         \[\leadsto \left(\left(-y\right) \cdot x + z \cdot x\right) + x \]

      4. +-commutative N/A

         \[\leadsto \left(z \cdot x + \left(-y\right) \cdot x\right) + x \]

      5. lift-*.f64 N/A

         \[\leadsto \left(z \cdot x + \left(-y\right) \cdot x\right) + x \]

      6. distribute-rgt-out N/A

         \[\leadsto x \cdot \left(z + \left(-y\right)\right) + x \]

      7. lift-neg.f64 N/A

         \[\leadsto x \cdot \left(z + \left(\mathsf{neg}\left(y\right)\right)\right) + x \]

      8. sub-flip N/A

         \[\leadsto x \cdot \left(z - y\right) + x \]

      9. *-commutative N/A

         \[\leadsto \left(z - y\right) \cdot x + x \]

      10. lower-fma.f64 N/A

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

      11. lower--.f64 56.3%

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

   10. Applied rewrites 56.3%

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

Alternative 5: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := z \cdot \left(x - t\right)\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-133}:\\ \;\;\;\;x + t \cdot y\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+32}:\\ \;\;\;\;y \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- x t))))
   (if (<= z -2.05e-20)
     t_1
     (if (<= z 9.6e-133)
       (+ x (* t y))
       (if (<= z 2.5e+32) (* y (- t x)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double tmp;
	if (z <= -2.05e-20) {
		tmp = t_1;
	} else if (z <= 9.6e-133) {
		tmp = x + (t * y);
	} else if (z <= 2.5e+32) {
		tmp = y * (t - x);
	} else {
		tmp = t_1;
	}
	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)
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) :: t_1
    real(8) :: tmp
    t_1 = z * (x - t)
    if (z <= (-2.05d-20)) then
        tmp = t_1
    else if (z <= 9.6d-133) then
        tmp = x + (t * y)
    else if (z <= 2.5d+32) then
        tmp = y * (t - x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double tmp;
	if (z <= -2.05e-20) {
		tmp = t_1;
	} else if (z <= 9.6e-133) {
		tmp = x + (t * y);
	} else if (z <= 2.5e+32) {
		tmp = y * (t - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (x - t)
	tmp = 0
	if z <= -2.05e-20:
		tmp = t_1
	elif z <= 9.6e-133:
		tmp = x + (t * y)
	elif z <= 2.5e+32:
		tmp = y * (t - x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(x - t))
	tmp = 0.0
	if (z <= -2.05e-20)
		tmp = t_1;
	elseif (z <= 9.6e-133)
		tmp = Float64(x + Float64(t * y));
	elseif (z <= 2.5e+32)
		tmp = Float64(y * Float64(t - x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (x - t);
	tmp = 0.0;
	if (z <= -2.05e-20)
		tmp = t_1;
	elseif (z <= 9.6e-133)
		tmp = x + (t * y);
	elseif (z <= 2.5e+32)
		tmp = y * (t - x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.05e-20], t$95$1, If[LessEqual[z, 9.6e-133], N[(x + N[(t * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5e+32], N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.05e-20 or 2.4999999999999999e32 < z

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 45.2%

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

   6. Applied rewrites 45.2%

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

   if -2.05e-20 < z < 9.6e-133

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{y \cdot \left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 60.3%

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto x + t \cdot \color{blue}{y} \]
   6. Step-by-step derivation

      1. lower-*.f64 42.1%

         \[\leadsto x + t \cdot y \]

   7. Applied rewrites 42.1%

      \[\leadsto x + t \cdot \color{blue}{y} \]

   if 9.6e-133 < z < 2.4999999999999999e32

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 64.8%

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(t - x\right)} \]

      2. lower--.f64 44.7%

         \[\leadsto y \cdot \left(t - \color{blue}{x}\right) \]

   9. Applied rewrites 44.7%

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;y \leq 420000000000:\\ \;\;\;\;z \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- t x))))
   (if (<= y -7.2e+29)
     t_1
     (if (<= y 7.2e-218)
       (fma z x x)
       (if (<= y 420000000000.0) (* z (- x t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double tmp;
	if (y <= -7.2e+29) {
		tmp = t_1;
	} else if (y <= 7.2e-218) {
		tmp = fma(z, x, x);
	} else if (y <= 420000000000.0) {
		tmp = z * (x - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	tmp = 0.0
	if (y <= -7.2e+29)
		tmp = t_1;
	elseif (y <= 7.2e-218)
		tmp = fma(z, x, x);
	elseif (y <= 420000000000.0)
		tmp = Float64(z * Float64(x - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e+29], t$95$1, If[LessEqual[y, 7.2e-218], N[(z * x + x), $MachinePrecision], If[LessEqual[y, 420000000000.0], N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.1999999999999995e29 or 4.2e11 < y

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 64.8%

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(t - x\right)} \]

      2. lower--.f64 44.7%

         \[\leadsto y \cdot \left(t - \color{blue}{x}\right) \]

   9. Applied rewrites 44.7%

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]

   if -7.1999999999999995e29 < y < 7.2000000000000002e-218

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 56.3%

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

   6. Applied rewrites 56.3%

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

   7. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + x \cdot \color{blue}{z} \]

      2. lower-*.f64 37.9%

         \[\leadsto x + x \cdot z \]

   9. Applied rewrites 37.9%

      \[\leadsto x + \color{blue}{x \cdot z} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto x + x \cdot \color{blue}{z} \]

       2. lift-*.f64 N/A

          \[\leadsto x + x \cdot z \]

       3. *-commutative N/A

          \[\leadsto x + z \cdot x \]

       4. distribute-rgt1-in N/A

          \[\leadsto \left(z + 1\right) \cdot x \]

       5. distribute-lft1-in N/A

          \[\leadsto z \cdot x + x \]

       6. lower-fma.f64 38.0%

          \[\leadsto \mathsf{fma}\left(z, x, x\right) \]

   11. Applied rewrites 38.0%

       \[\leadsto \mathsf{fma}\left(z, x, x\right) \]

   if 7.2000000000000002e-218 < y < 4.2e11

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 45.2%

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

   6. Applied rewrites 45.2%

      \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 420000000000:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- t x))))
   (if (<= y -7.2e+29) t_1 (if (<= y 420000000000.0) (fma z x x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double tmp;
	if (y <= -7.2e+29) {
		tmp = t_1;
	} else if (y <= 420000000000.0) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	tmp = 0.0
	if (y <= -7.2e+29)
		tmp = t_1;
	elseif (y <= 420000000000.0)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e+29], t$95$1, If[LessEqual[y, 420000000000.0], N[(z * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.1999999999999995e29 or 4.2e11 < y

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 64.8%

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(t - x\right)} \]

      2. lower--.f64 44.7%

         \[\leadsto y \cdot \left(t - \color{blue}{x}\right) \]

   9. Applied rewrites 44.7%

      \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]

   if -7.1999999999999995e29 < y < 4.2e11

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 56.3%

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

   6. Applied rewrites 56.3%

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

   7. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + x \cdot \color{blue}{z} \]

      2. lower-*.f64 37.9%

         \[\leadsto x + x \cdot z \]

   9. Applied rewrites 37.9%

      \[\leadsto x + \color{blue}{x \cdot z} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto x + x \cdot \color{blue}{z} \]

       2. lift-*.f64 N/A

          \[\leadsto x + x \cdot z \]

       3. *-commutative N/A

          \[\leadsto x + z \cdot x \]

       4. distribute-rgt1-in N/A

          \[\leadsto \left(z + 1\right) \cdot x \]

       5. distribute-lft1-in N/A

          \[\leadsto z \cdot x + x \]

       6. lower-fma.f64 38.0%

          \[\leadsto \mathsf{fma}\left(z, x, x\right) \]

   11. Applied rewrites 38.0%

       \[\leadsto \mathsf{fma}\left(z, x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := t \cdot \left(y - z\right)\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (- y z))))
   (if (<= t -4.4e+92) t_1 (if (<= t 4.4e+42) (fma z x x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t * (y - z);
	double tmp;
	if (t <= -4.4e+92) {
		tmp = t_1;
	} else if (t <= 4.4e+42) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(t * Float64(y - z))
	tmp = 0.0
	if (t <= -4.4e+92)
		tmp = t_1;
	elseif (t <= 4.4e+42)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.4e+92], t$95$1, If[LessEqual[t, 4.4e+42], N[(z * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.3999999999999998e92 or 4.4000000000000003e42 < t

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 64.8%

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{t \cdot \left(y - z\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 49.3%

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

   9. Applied rewrites 49.3%

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

   if -4.3999999999999998e92 < t < 4.4000000000000003e42

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 56.3%

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

   6. Applied rewrites 56.3%

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

   7. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + x \cdot \color{blue}{z} \]

      2. lower-*.f64 37.9%

         \[\leadsto x + x \cdot z \]

   9. Applied rewrites 37.9%

      \[\leadsto x + \color{blue}{x \cdot z} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto x + x \cdot \color{blue}{z} \]

       2. lift-*.f64 N/A

          \[\leadsto x + x \cdot z \]

       3. *-commutative N/A

          \[\leadsto x + z \cdot x \]

       4. distribute-rgt1-in N/A

          \[\leadsto \left(z + 1\right) \cdot x \]

       5. distribute-lft1-in N/A

          \[\leadsto z \cdot x + x \]

       6. lower-fma.f64 38.0%

          \[\leadsto \mathsf{fma}\left(z, x, x\right) \]

   11. Applied rewrites 38.0%

       \[\leadsto \mathsf{fma}\left(z, x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 46.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \left(-t\right) \cdot z\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t) z)))
   (if (<= t -4.4e+92) t_1 (if (<= t 2.9e+151) (fma z x x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -t * z;
	double tmp;
	if (t <= -4.4e+92) {
		tmp = t_1;
	} else if (t <= 2.9e+151) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-t) * z)
	tmp = 0.0
	if (t <= -4.4e+92)
		tmp = t_1;
	elseif (t <= 2.9e+151)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-t) * z), $MachinePrecision]}, If[LessEqual[t, -4.4e+92], t$95$1, If[LessEqual[t, 2.9e+151], N[(z * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.3999999999999998e92 or 2.9000000000000002e151 < t

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 45.2%

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

   6. Applied rewrites 45.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto z \cdot \left(-1 \cdot \color{blue}{t}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 27.2%

         \[\leadsto z \cdot \left(-1 \cdot t\right) \]

   9. Applied rewrites 27.2%

      \[\leadsto z \cdot \left(-1 \cdot \color{blue}{t}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 27.2%

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

       4. lift-*.f64 N/A

          \[\leadsto \left(-1 \cdot t\right) \cdot z \]

       5. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(t\right)\right) \cdot z \]

       6. lower-neg.f64 27.2%

          \[\leadsto \left(-t\right) \cdot z \]

   11. Applied rewrites 27.2%

       \[\leadsto \left(-t\right) \cdot \color{blue}{z} \]

   if -4.3999999999999998e92 < t < 2.9000000000000002e151

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

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

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

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

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 56.3%

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

   6. Applied rewrites 56.3%

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

   7. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{x \cdot z} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto x + x \cdot \color{blue}{z} \]

      2. lower-*.f64 37.9%

         \[\leadsto x + x \cdot z \]

   9. Applied rewrites 37.9%

      \[\leadsto x + \color{blue}{x \cdot z} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto x + x \cdot \color{blue}{z} \]

       2. lift-*.f64 N/A

          \[\leadsto x + x \cdot z \]

       3. *-commutative N/A

          \[\leadsto x + z \cdot x \]

       4. distribute-rgt1-in N/A

          \[\leadsto \left(z + 1\right) \cdot x \]

       5. distribute-lft1-in N/A

          \[\leadsto z \cdot x + x \]

       6. lower-fma.f64 38.0%

          \[\leadsto \mathsf{fma}\left(z, x, x\right) \]

   11. Applied rewrites 38.0%

       \[\leadsto \mathsf{fma}\left(z, x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 27.2% accurate, 2.4× speedup

Math

\[\left(-t\right) \cdot z \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[((-t) * z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

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

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

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

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 45.2%

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

6. Applied rewrites 45.2%

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

7. Taylor expanded in x around 0

   \[\leadsto z \cdot \left(-1 \cdot \color{blue}{t}\right) \]
8. Step-by-step derivation

   1. lower-*.f64 27.2%

      \[\leadsto z \cdot \left(-1 \cdot t\right) \]

9. Applied rewrites 27.2%

   \[\leadsto z \cdot \left(-1 \cdot \color{blue}{t}\right) \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 27.2%

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

    4. lift-*.f64 N/A

       \[\leadsto \left(-1 \cdot t\right) \cdot z \]

    5. mul-1-neg N/A

       \[\leadsto \left(\mathsf{neg}\left(t\right)\right) \cdot z \]

    6. lower-neg.f64 27.2%

       \[\leadsto \left(-t\right) \cdot z \]

11. Applied rewrites 27.2%

    \[\leadsto \left(-t\right) \cdot \color{blue}{z} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025205 
(FPCore (x y z t)
  :name "Data.Metrics.Snapshot:quantile from metrics-0.3.0.2"
  :precision binary64
  (+ x (* (- y z) (- t x))))