Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.4% → 97.8%

Time: 8.4s

Alternatives: 12

Speedup: 1.0×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y - x}{\frac{t}{z}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - x) / (t / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. associate-/l* 96.9%

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

3. Simplified 96.9%

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

   1. clear-num 96.9%

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

   2. un-div-inv 97.1%

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

6. Applied egg-rr 97.1%

   \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]
7. Add Preprocessing

Alternative 2: 49.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{+104}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{+40} \lor \neg \left(t \leq -2.8 \cdot 10^{-65}\right) \land t \leq 10.5:\\ \;\;\;\;x \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -8.2e+104)
   x
   (if (or (<= t -2.5e+40) (and (not (<= t -2.8e-65)) (<= t 10.5)))
     (* x (/ z (- t)))
     x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.2e+104) {
		tmp = x;
	} else if ((t <= -2.5e+40) || (!(t <= -2.8e-65) && (t <= 10.5))) {
		tmp = x * (z / -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-8.2d+104)) then
        tmp = x
    else if ((t <= (-2.5d+40)) .or. (.not. (t <= (-2.8d-65))) .and. (t <= 10.5d0)) then
        tmp = x * (z / -t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.2e+104) {
		tmp = x;
	} else if ((t <= -2.5e+40) || (!(t <= -2.8e-65) && (t <= 10.5))) {
		tmp = x * (z / -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -8.2e+104:
		tmp = x
	elif (t <= -2.5e+40) or (not (t <= -2.8e-65) and (t <= 10.5)):
		tmp = x * (z / -t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8.2e+104)
		tmp = x;
	elseif ((t <= -2.5e+40) || (!(t <= -2.8e-65) && (t <= 10.5)))
		tmp = Float64(x * Float64(z / Float64(-t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -8.2e+104)
		tmp = x;
	elseif ((t <= -2.5e+40) || (~((t <= -2.8e-65)) && (t <= 10.5)))
		tmp = x * (z / -t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -8.2e+104], x, If[Or[LessEqual[t, -2.5e+40], And[N[Not[LessEqual[t, -2.8e-65]], $MachinePrecision], LessEqual[t, 10.5]]], N[(x * N[(z / (-t)), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -8.1999999999999997e104 or -2.50000000000000002e40 < t < -2.8e-65 or 10.5 < t

   1. Initial program 89.9%

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

      1. +-commutative 89.9%

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

      2. associate-/l* 97.7%

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

      3. fma-define 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in z around 0 65.9%

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

   if -8.1999999999999997e104 < t < -2.50000000000000002e40 or -2.8e-65 < t < 10.5

   1. Initial program 98.3%

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

      1. +-commutative 98.3%

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

      2. associate-/l* 96.0%

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

      3. fma-define 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in y around 0 59.3%

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

      1. mul-1-neg 59.3%

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

      2. unsub-neg 59.3%

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

      3. *-commutative 59.3%

         \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

      4. associate-*l/ 59.9%

         \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

      5. cancel-sign-sub-inv 59.9%

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

      6. mul-1-neg 59.9%

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

      7. *-lft-identity 59.9%

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

      8. distribute-rgt-in 59.9%

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

      9. mul-1-neg 59.9%

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

      10. unsub-neg 59.9%

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

   7. Simplified 59.9%

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

   8. Taylor expanded in z around inf 50.1%

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

      1. mul-1-neg 50.1%

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

      2. distribute-frac-neg2 50.1%

         \[\leadsto x \cdot \color{blue}{\frac{z}{-t}} \]

   10. Simplified 50.1%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{+104}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{+40} \lor \neg \left(t \leq -2.8 \cdot 10^{-65}\right) \land t \leq 10.5:\\ \;\;\;\;x \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{+40}:\\ \;\;\;\;z \cdot \frac{x}{-t}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-65}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 88000:\\ \;\;\;\;x \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.25e+105)
   x
   (if (<= t -1.9e+40)
     (* z (/ x (- t)))
     (if (<= t -2.8e-65) x (if (<= t 88000.0) (* x (/ z (- t))) x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.25e+105) {
		tmp = x;
	} else if (t <= -1.9e+40) {
		tmp = z * (x / -t);
	} else if (t <= -2.8e-65) {
		tmp = x;
	} else if (t <= 88000.0) {
		tmp = x * (z / -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.25d+105)) then
        tmp = x
    else if (t <= (-1.9d+40)) then
        tmp = z * (x / -t)
    else if (t <= (-2.8d-65)) then
        tmp = x
    else if (t <= 88000.0d0) then
        tmp = x * (z / -t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.25e+105) {
		tmp = x;
	} else if (t <= -1.9e+40) {
		tmp = z * (x / -t);
	} else if (t <= -2.8e-65) {
		tmp = x;
	} else if (t <= 88000.0) {
		tmp = x * (z / -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.25e+105:
		tmp = x
	elif t <= -1.9e+40:
		tmp = z * (x / -t)
	elif t <= -2.8e-65:
		tmp = x
	elif t <= 88000.0:
		tmp = x * (z / -t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.25e+105)
		tmp = x;
	elseif (t <= -1.9e+40)
		tmp = Float64(z * Float64(x / Float64(-t)));
	elseif (t <= -2.8e-65)
		tmp = x;
	elseif (t <= 88000.0)
		tmp = Float64(x * Float64(z / Float64(-t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.25e+105)
		tmp = x;
	elseif (t <= -1.9e+40)
		tmp = z * (x / -t);
	elseif (t <= -2.8e-65)
		tmp = x;
	elseif (t <= 88000.0)
		tmp = x * (z / -t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.25e+105], x, If[LessEqual[t, -1.9e+40], N[(z * N[(x / (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.8e-65], x, If[LessEqual[t, 88000.0], N[(x * N[(z / (-t)), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.25000000000000011e105 or -1.90000000000000002e40 < t < -2.8e-65 or 88000 < t

   1. Initial program 89.9%

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

      1. +-commutative 89.9%

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

      2. associate-/l* 97.7%

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

      3. fma-define 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in z around 0 65.9%

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

   if -1.25000000000000011e105 < t < -1.90000000000000002e40

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-/l* 91.6%

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

      3. fma-define 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in y around 0 59.7%

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

      1. mul-1-neg 59.7%

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

      2. unsub-neg 59.7%

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

      3. *-commutative 59.7%

         \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

      4. associate-*l/ 59.9%

         \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

      5. cancel-sign-sub-inv 59.9%

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

      6. mul-1-neg 59.9%

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

      7. *-lft-identity 59.9%

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

      8. distribute-rgt-in 60.0%

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

      9. mul-1-neg 60.0%

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

      10. unsub-neg 60.0%

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

   7. Simplified 60.0%

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

   8. Taylor expanded in z around inf 45.7%

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

      1. mul-1-neg 45.7%

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

      2. distribute-frac-neg2 45.7%

         \[\leadsto x \cdot \color{blue}{\frac{z}{-t}} \]

   10. Simplified 45.7%

       \[\leadsto x \cdot \color{blue}{\frac{z}{-t}} \]

   11. Taylor expanded in x around 0 45.6%

       \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t}} \]
   12. Step-by-step derivation

       1. *-commutative 45.6%

          \[\leadsto -1 \cdot \frac{\color{blue}{z \cdot x}}{t} \]

       2. associate-/l* 45.7%

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

       3. associate-*r* 45.7%

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

       4. neg-mul-1 45.7%

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

   13. Simplified 45.7%

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

   if -2.8e-65 < t < 88000

   1. Initial program 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-/l* 96.4%

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

      3. fma-define 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in y around 0 59.2%

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

      1. mul-1-neg 59.2%

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

      2. unsub-neg 59.2%

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

      3. *-commutative 59.2%

         \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

      4. associate-*l/ 59.9%

         \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

      5. cancel-sign-sub-inv 59.9%

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

      6. mul-1-neg 59.9%

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

      7. *-lft-identity 59.9%

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

      8. distribute-rgt-in 59.9%

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

      9. mul-1-neg 59.9%

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

      10. unsub-neg 59.9%

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

   7. Simplified 59.9%

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

   8. Taylor expanded in z around inf 50.5%

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

      1. mul-1-neg 50.5%

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

      2. distribute-frac-neg2 50.5%

         \[\leadsto x \cdot \color{blue}{\frac{z}{-t}} \]

   10. Simplified 50.5%

       \[\leadsto x \cdot \color{blue}{\frac{z}{-t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{+40}:\\ \;\;\;\;z \cdot \frac{x}{-t}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-65}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 88000:\\ \;\;\;\;x \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-68} \lor \neg \left(z \leq 1.1 \cdot 10^{-162}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.9e-68) (not (<= z 1.1e-162)))
   (+ x (* z (/ (- y x) t)))
   (+ x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.9e-68) || !(z <= 1.1e-162)) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = x + ((y * z) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.9d-68)) .or. (.not. (z <= 1.1d-162))) then
        tmp = x + (z * ((y - x) / t))
    else
        tmp = x + ((y * z) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.9e-68) || !(z <= 1.1e-162)) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = x + ((y * z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.9e-68) or not (z <= 1.1e-162):
		tmp = x + (z * ((y - x) / t))
	else:
		tmp = x + ((y * z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.9e-68) || !(z <= 1.1e-162))
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / t)));
	else
		tmp = Float64(x + Float64(Float64(y * z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.9e-68) || ~((z <= 1.1e-162)))
		tmp = x + (z * ((y - x) / t));
	else
		tmp = x + ((y * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.9e-68], N[Not[LessEqual[z, 1.1e-162]], $MachinePrecision]], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.90000000000000019e-68 or 1.1e-162 < z

   1. Initial program 92.5%

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

      1. associate-/l* 97.2%

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

   3. Simplified 97.2%

      \[\leadsto \color{blue}{x + \left(y - x\right) \cdot \frac{z}{t}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 84.6%

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

      1. associate-/l* 87.2%

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

      2. associate-*r* 87.2%

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

      3. neg-mul-1 87.2%

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

      4. associate-*r/ 87.6%

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

      5. distribute-rgt-out 97.2%

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

      6. +-commutative 97.2%

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

      7. sub-neg 97.2%

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

      8. associate-*l/ 92.5%

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

      9. associate-*r/ 98.4%

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

   7. Simplified 98.4%

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

   if -1.90000000000000019e-68 < z < 1.1e-162

   1. Initial program 97.4%

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

   3. Taylor expanded in y around inf 93.7%

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

      1. *-commutative 93.7%

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

   5. Simplified 93.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-68} \lor \neg \left(z \leq 1.1 \cdot 10^{-162}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{t - z}{t}\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.45e-57)
   (* x (/ (- t z) t))
   (if (<= x 1.76e-32) (+ x (/ (* y z) t)) (* x (- 1.0 (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.45e-57) {
		tmp = x * ((t - z) / t);
	} else if (x <= 1.76e-32) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-1.45d-57)) then
        tmp = x * ((t - z) / t)
    else if (x <= 1.76d-32) then
        tmp = x + ((y * z) / t)
    else
        tmp = x * (1.0d0 - (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.45e-57) {
		tmp = x * ((t - z) / t);
	} else if (x <= 1.76e-32) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.45e-57:
		tmp = x * ((t - z) / t)
	elif x <= 1.76e-32:
		tmp = x + ((y * z) / t)
	else:
		tmp = x * (1.0 - (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.45e-57)
		tmp = Float64(x * Float64(Float64(t - z) / t));
	elseif (x <= 1.76e-32)
		tmp = Float64(x + Float64(Float64(y * z) / t));
	else
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.45e-57)
		tmp = x * ((t - z) / t);
	elseif (x <= 1.76e-32)
		tmp = x + ((y * z) / t);
	else
		tmp = x * (1.0 - (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.45e-57], N[(x * N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.76e-32], N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45000000000000013e-57

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 84.0%

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

      1. mul-1-neg 84.0%

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

      2. unsub-neg 84.0%

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

      3. *-commutative 84.0%

         \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

      4. associate-*l/ 90.6%

         \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

      5. cancel-sign-sub-inv 90.6%

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

      6. mul-1-neg 90.6%

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

      7. *-lft-identity 90.6%

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

      8. distribute-rgt-in 90.6%

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

      9. mul-1-neg 90.6%

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

      10. unsub-neg 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in t around 0 90.7%

      \[\leadsto x \cdot \color{blue}{\frac{t - z}{t}} \]

   if -1.45000000000000013e-57 < x < 1.76000000000000004e-32

   1. Initial program 97.7%

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

   3. Taylor expanded in y around inf 84.5%

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

      1. *-commutative 84.5%

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

   5. Simplified 84.5%

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

   if 1.76000000000000004e-32 < x

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. unsub-neg 85.5%

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

      3. *-commutative 85.5%

         \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

      4. associate-*l/ 89.2%

         \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

      5. cancel-sign-sub-inv 89.2%

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

      6. mul-1-neg 89.2%

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

      7. *-lft-identity 89.2%

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

      8. distribute-rgt-in 89.2%

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

      9. mul-1-neg 89.2%

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

      10. unsub-neg 89.2%

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

   7. Simplified 89.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{t - z}{t}\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{-32}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{t - z}{t}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{-33}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.45e-57)
   (* x (/ (- t z) t))
   (if (<= x 2.85e-33) (+ x (/ y (/ t z))) (* x (- 1.0 (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.45e-57) {
		tmp = x * ((t - z) / t);
	} else if (x <= 2.85e-33) {
		tmp = x + (y / (t / z));
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-1.45d-57)) then
        tmp = x * ((t - z) / t)
    else if (x <= 2.85d-33) then
        tmp = x + (y / (t / z))
    else
        tmp = x * (1.0d0 - (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.45e-57) {
		tmp = x * ((t - z) / t);
	} else if (x <= 2.85e-33) {
		tmp = x + (y / (t / z));
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.45e-57:
		tmp = x * ((t - z) / t)
	elif x <= 2.85e-33:
		tmp = x + (y / (t / z))
	else:
		tmp = x * (1.0 - (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.45e-57)
		tmp = Float64(x * Float64(Float64(t - z) / t));
	elseif (x <= 2.85e-33)
		tmp = Float64(x + Float64(y / Float64(t / z)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.45e-57)
		tmp = x * ((t - z) / t);
	elseif (x <= 2.85e-33)
		tmp = x + (y / (t / z));
	else
		tmp = x * (1.0 - (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.45e-57], N[(x * N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.85e-33], N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45000000000000013e-57

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 84.0%

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

      1. mul-1-neg 84.0%

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

      2. unsub-neg 84.0%

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

      3. *-commutative 84.0%

         \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

      4. associate-*l/ 90.6%

         \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

      5. cancel-sign-sub-inv 90.6%

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

      6. mul-1-neg 90.6%

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

      7. *-lft-identity 90.6%

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

      8. distribute-rgt-in 90.6%

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

      9. mul-1-neg 90.6%

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

      10. unsub-neg 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in t around 0 90.7%

      \[\leadsto x \cdot \color{blue}{\frac{t - z}{t}} \]

   if -1.45000000000000013e-57 < x < 2.85000000000000013e-33

   1. Initial program 97.7%

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

      1. associate-/l* 91.5%

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

   3. Simplified 91.5%

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

      1. clear-num 91.4%

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

      2. un-div-inv 92.1%

         \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

   6. Applied egg-rr 92.1%

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

   7. Taylor expanded in y around inf 84.5%

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

      1. associate-*l/ 80.3%

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

      2. associate-/r/ 80.7%

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

   9. Simplified 80.7%

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

   if 2.85000000000000013e-33 < x

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. unsub-neg 85.5%

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

      3. *-commutative 85.5%

         \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

      4. associate-*l/ 89.2%

         \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

      5. cancel-sign-sub-inv 89.2%

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

      6. mul-1-neg 89.2%

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

      7. *-lft-identity 89.2%

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

      8. distribute-rgt-in 89.2%

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

      9. mul-1-neg 89.2%

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

      10. unsub-neg 89.2%

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

   7. Simplified 89.2%

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

Alternative 7: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{t - z}{t}\\ \mathbf{elif}\;x \leq 3.75 \cdot 10^{-32}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.05e-57)
   (* x (/ (- t z) t))
   (if (<= x 3.75e-32) (+ x (* z (/ y t))) (* x (- 1.0 (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.05e-57) {
		tmp = x * ((t - z) / t);
	} else if (x <= 3.75e-32) {
		tmp = x + (z * (y / t));
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-1.05d-57)) then
        tmp = x * ((t - z) / t)
    else if (x <= 3.75d-32) then
        tmp = x + (z * (y / t))
    else
        tmp = x * (1.0d0 - (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.05e-57) {
		tmp = x * ((t - z) / t);
	} else if (x <= 3.75e-32) {
		tmp = x + (z * (y / t));
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.05e-57:
		tmp = x * ((t - z) / t)
	elif x <= 3.75e-32:
		tmp = x + (z * (y / t))
	else:
		tmp = x * (1.0 - (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.05e-57)
		tmp = Float64(x * Float64(Float64(t - z) / t));
	elseif (x <= 3.75e-32)
		tmp = Float64(x + Float64(z * Float64(y / t)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.05e-57)
		tmp = x * ((t - z) / t);
	elseif (x <= 3.75e-32)
		tmp = x + (z * (y / t));
	else
		tmp = x * (1.0 - (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.05e-57], N[(x * N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.75e-32], N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e-57

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 84.0%

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

      1. mul-1-neg 84.0%

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

      2. unsub-neg 84.0%

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

      3. *-commutative 84.0%

         \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

      4. associate-*l/ 90.6%

         \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

      5. cancel-sign-sub-inv 90.6%

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

      6. mul-1-neg 90.6%

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

      7. *-lft-identity 90.6%

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

      8. distribute-rgt-in 90.6%

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

      9. mul-1-neg 90.6%

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

      10. unsub-neg 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in t around 0 90.7%

      \[\leadsto x \cdot \color{blue}{\frac{t - z}{t}} \]

   if -1.05e-57 < x < 3.74999999999999977e-32

   1. Initial program 97.7%

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

      1. associate-/l* 91.5%

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

   3. Simplified 91.5%

      \[\leadsto \color{blue}{x + \left(y - x\right) \cdot \frac{z}{t}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-/l* 80.3%

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

   7. Simplified 80.3%

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

   if 3.74999999999999977e-32 < x

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. unsub-neg 85.5%

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

      3. *-commutative 85.5%

         \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

      4. associate-*l/ 89.2%

         \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

      5. cancel-sign-sub-inv 89.2%

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

      6. mul-1-neg 89.2%

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

      7. *-lft-identity 89.2%

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

      8. distribute-rgt-in 89.2%

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

      9. mul-1-neg 89.2%

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

      10. unsub-neg 89.2%

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

   7. Simplified 89.2%

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

Alternative 8: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-57}:\\ \;\;\;\;x \cdot \frac{t - z}{t}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-36}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.1e-57)
   (* x (/ (- t z) t))
   (if (<= x 2.6e-36) (+ x (* y (/ z t))) (* x (- 1.0 (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e-57) {
		tmp = x * ((t - z) / t);
	} else if (x <= 2.6e-36) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-1.1d-57)) then
        tmp = x * ((t - z) / t)
    else if (x <= 2.6d-36) then
        tmp = x + (y * (z / t))
    else
        tmp = x * (1.0d0 - (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.1e-57) {
		tmp = x * ((t - z) / t);
	} else if (x <= 2.6e-36) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.1e-57:
		tmp = x * ((t - z) / t)
	elif x <= 2.6e-36:
		tmp = x + (y * (z / t))
	else:
		tmp = x * (1.0 - (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.1e-57)
		tmp = Float64(x * Float64(Float64(t - z) / t));
	elseif (x <= 2.6e-36)
		tmp = Float64(x + Float64(y * Float64(z / t)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.1e-57)
		tmp = x * ((t - z) / t);
	elseif (x <= 2.6e-36)
		tmp = x + (y * (z / t));
	else
		tmp = x * (1.0 - (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.1e-57], N[(x * N[(N[(t - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e-36], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.09999999999999999e-57

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 84.0%

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

      1. mul-1-neg 84.0%

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

      2. unsub-neg 84.0%

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

      3. *-commutative 84.0%

         \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

      4. associate-*l/ 90.6%

         \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

      5. cancel-sign-sub-inv 90.6%

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

      6. mul-1-neg 90.6%

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

      7. *-lft-identity 90.6%

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

      8. distribute-rgt-in 90.6%

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

      9. mul-1-neg 90.6%

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

      10. unsub-neg 90.6%

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

   7. Simplified 90.6%

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

   8. Taylor expanded in t around 0 90.7%

      \[\leadsto x \cdot \color{blue}{\frac{t - z}{t}} \]

   if -1.09999999999999999e-57 < x < 2.6e-36

   1. Initial program 97.7%

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

      1. associate-/l* 91.5%

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

   3. Simplified 91.5%

      \[\leadsto \color{blue}{x + \left(y - x\right) \cdot \frac{z}{t}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 84.5%

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

      1. associate-*r/ 80.1%

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

   7. Simplified 80.1%

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

   if 2.6e-36 < x

   1. Initial program 92.8%

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

      1. +-commutative 92.8%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 85.5%

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

      1. mul-1-neg 85.5%

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

      2. unsub-neg 85.5%

         \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

      3. *-commutative 85.5%

         \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

      4. associate-*l/ 89.2%

         \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

      5. cancel-sign-sub-inv 89.2%

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

      6. mul-1-neg 89.2%

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

      7. *-lft-identity 89.2%

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

      8. distribute-rgt-in 89.2%

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

      9. mul-1-neg 89.2%

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

      10. unsub-neg 89.2%

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

   7. Simplified 89.2%

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

Alternative 9: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. associate-/l* 96.9%

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

3. Simplified 96.9%

   \[\leadsto \color{blue}{x + \left(y - x\right) \cdot \frac{z}{t}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 10: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{t - z}{t} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. +-commutative 94.0%

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

   2. associate-/l* 96.9%

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

   3. fma-define 96.9%

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

3. Simplified 96.9%

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

5. Taylor expanded in y around 0 65.1%

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

   1. mul-1-neg 65.1%

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

   2. unsub-neg 65.1%

      \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

   3. *-commutative 65.1%

      \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

   4. associate-*l/ 68.7%

      \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

   5. cancel-sign-sub-inv 68.7%

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

   6. mul-1-neg 68.7%

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

   7. *-lft-identity 68.7%

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

   8. distribute-rgt-in 68.7%

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

   9. mul-1-neg 68.7%

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

   10. unsub-neg 68.7%

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

7. Simplified 68.7%

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

8. Taylor expanded in t around 0 68.7%

   \[\leadsto x \cdot \color{blue}{\frac{t - z}{t}} \]
9. Add Preprocessing

Alternative 11: 64.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. +-commutative 94.0%

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

   2. associate-/l* 96.9%

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

   3. fma-define 96.9%

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

3. Simplified 96.9%

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

5. Taylor expanded in y around 0 65.1%

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

   1. mul-1-neg 65.1%

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

   2. unsub-neg 65.1%

      \[\leadsto \color{blue}{x - \frac{x \cdot z}{t}} \]

   3. *-commutative 65.1%

      \[\leadsto x - \frac{\color{blue}{z \cdot x}}{t} \]

   4. associate-*l/ 68.7%

      \[\leadsto x - \color{blue}{\frac{z}{t} \cdot x} \]

   5. cancel-sign-sub-inv 68.7%

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

   6. mul-1-neg 68.7%

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

   7. *-lft-identity 68.7%

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

   8. distribute-rgt-in 68.7%

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

   9. mul-1-neg 68.7%

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

   10. unsub-neg 68.7%

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

7. Simplified 68.7%

   \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{t}\right)} \]
8. Add Preprocessing

Alternative 12: 38.0% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

   1. +-commutative 94.0%

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

   2. associate-/l* 96.9%

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

   3. fma-define 96.9%

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

3. Simplified 96.9%

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

5. Taylor expanded in z around 0 39.8%

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

Developer target: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< x -9.025511195533005e-135)
   (- x (* (/ z t) (- x y)))
   (if (< x 4.275032163700715e-250)
     (+ x (* (/ (- y x) t) z))
     (+ x (/ (- y x) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x < (-9.025511195533005d-135)) then
        tmp = x - ((z / t) * (x - y))
    else if (x < 4.275032163700715d-250) then
        tmp = x + (((y - x) / t) * z)
    else
        tmp = x + ((y - x) / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x < -9.025511195533005e-135:
		tmp = x - ((z / t) * (x - y))
	elif x < 4.275032163700715e-250:
		tmp = x + (((y - x) / t) * z)
	else:
		tmp = x + ((y - x) / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x < -9.025511195533005e-135)
		tmp = Float64(x - Float64(Float64(z / t) * Float64(x - y)));
	elseif (x < 4.275032163700715e-250)
		tmp = Float64(x + Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x < -9.025511195533005e-135)
		tmp = x - ((z / t) * (x - y));
	elseif (x < 4.275032163700715e-250)
		tmp = x + (((y - x) / t) * z);
	else
		tmp = x + ((y - x) / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[x, -9.025511195533005e-135], N[(x - N[(N[(z / t), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[x, 4.275032163700715e-250], N[(x + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024086 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :alt
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

  (+ x (/ (* (- y x) z) t)))